Problem: Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{r^2 - 10r + 21}{r^2 - 3r}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 - 10r + 21}{r^2 - 3r} = \dfrac{(r - 7)(r - 3)}{(r)(r - 3)} $ Notice that the term $(r - 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r - 3)$ gives: $t = \dfrac{r - 7}{r}$ Since we divided by $(r - 3)$, $r \neq 3$. $t = \dfrac{r - 7}{r}; \space r \neq 3$